N 38: Fill & Download for Free

By jove, here’s one I can answer.Acepromazine is a drug that is prescribed as a sedative in dogs, cats, and horses. It was developed in the mid-20th century as an anti-psychotic. During a clinical trial in 1958, it was discovered that there were serious side effects including epilepsy. Furthermore, it was discovered that only 23% of patients in the groups receiving the drug (n=38) saw any form of improvement in their symptoms of schizophrenia. The authors declared that the drug “has a limited value” in humans.I’m unable to find a source detailing why acepromazine was originally prescribed in animals.My dog received a prescription for acepromazine due to severe anxiety of fireworks. We found that the drug isn’t a heavy sedative and doesn’t quell any anxiety, so we don’t use it. What we also discovered, through my own carelessness, was that the same dose used in my 75-pound dog would have been used in my 9-pound cat had he needed it.Fortunately, he only suffered the humiliation of us taking a few photos of him in his intoxicated state.Sources:Acepromazine - WikipediaClinical Trial of Acepromazine Maleate in Chronic Schizophrenia

The first thing to note is that 342 = 19*18. So let the number be x.So, x = 342*n + 47x = 342*n + 38 + 9x = 19*(18n + 2) + 9Clearly, when divided by 19, the remainder is 9.

You couldn’t. Or perhaps I don’t know what you mean by “analytically”.For any particular value of [math]n[/math][math][/math], the task before you is a finite task: factor the number [math]10^n+1[/math]. It’s not necessarily an easy task, but it’s a finite one. There’s no mystery around how to do it, though nobody knows just how efficiently it can be done. Any algorithm for factorization, simple or clever, will provide the answer.Beyond that, however, there’s no slick formula or some especially short algorithmic description of the factors which relies on the particular form of those numbers. There are more efficient factorization algorithms for such “generalized Fermat numbers” than for arbitrary numbers of comparable size, but they are not more “analytic” than any other factorization method.Take [math]n=38[/math] as an example. You want to factor [math]10^{38}+1[/math]. Cool. Since [math]19[/math] is odd, you know that [math]100^{19}+1[/math] is divisible by [math]100+1[/math], because [math]X+1[/math] divides [math]X^m+1[/math] whenever [math]m[/math] is odd.So, [math]10^{38}+1[/math] has [math]101[/math] as a prime factor. Great: that was reasonably “analytic”. We used the structure of the number to quickly find a prime factor without working too hard.Now we still need to factor [math](10^{38}+1)/101[/math], which is a [math]35[/math]-digit number with a spiffy decimal notation:[math]990099009900990099009900990099009901[/math]And now, our analytical methods are dried up. This number isn’t prime: in fact it is[math]722817036322379041 \times 1369778187490592461[/math]but I can’t tell you “why” those are the factors. They don’t fall out of some general theory. They just happen to be there.For arbitrary values of [math]n[/math][math][/math], we can use the “odd power” trick to find some factors of [math]10^n+1[/math]. What’s left over is sometimes prime, sometimes a product of lots of little fragments, and sometimes a product of some monstrous primes you won’t be able to find with any reasonable technology, ever.We don’t even know if there are any primes of this form beyond [math]101[/math]. Due to the odd power trick they can only arise when [math]n[/math][math][/math] is a power of [math]2[/math], but we don’t know anything beyond that. Perhaps there are none. Perhaps there are infinitely many. Perhaps something in between. We don’t know.